Give a possible set of values of the four quantum numbers for the \(4 s\) and \(3 d\) electrons in titanium.

Titanium has an atomic number of 22, and its electron configuration can be written as: \[1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^2\] Now we can analyze the 4s and 3d electrons:

1. Principal quantum number (n): For the 4s orbital, the principal quantum number is \(n = 4\). 2. Angular momentum quantum number (l): For s orbitals, the angular momentum quantum number is \(l = 0\). 3. Magnetic quantum number (m_l): For l = 0, there is only one possible value of the magnetic quantum number, \(m_l = 0\). 4. Electron Spin quantum number (m_s): There are two possibilities for the electron spin quantum number: either \(m_s = +\frac{1}{2}\) or \(m_s = -\frac{1}{2}\). Therefore, the two possible sets of quantum numbers for 4s electrons in titanium are: - \((n, l, m_l, m_s) = (4, 0, 0, +\frac{1}{2})\) - \((n, l, m_l, m_s) = (4, 0, 0, -\frac{1}{2})\)

1. Principal quantum number (n): For the 3d orbital, the principal quantum number is \(n = 3\). 2. Angular momentum quantum number (l): For d orbitals, the angular momentum quantum number is \(l = 2\). 3. Magnetic quantum number (m_l): For l = 2, we have five possible values of the magnetic quantum number: \(-2, -1, 0, 1, 2\). However, the titanium atom has only two 3d electrons; therefore, m_l can have any two of the five possible values (we don't know which two values those are). 4. Electron Spin quantum number (m_s): There are two possibilities for the electron spin quantum number: either \(m_s = +\frac{1}{2}\) or \(m_s = -\frac{1}{2}\). So, for the 3d electrons, we have four possible sets of quantum numbers (ignoring the cases of 3 other m_l values): - \((n, l, m_l, m_s) = (3, 2, m_{l1}, +\frac{1}{2})\), where \(m_{l1}\) is one of \(-2, -1, 0, 1, 2\) - \((n, l, m_l, m_s) = (3, 2, m_{l1}, -\frac{1}{2})\), where \(m_{l1}\) is one of \(-2, -1, 0, 1, 2\) - \((n, l, m_l, m_s) = (3, 2, m_{l2}, +\frac{1}{2})\), where \(m_{l2}\) is one of \(-2, -1, 0, 1, 2\) and \(m_{l2} \neq m_{l1}\) - \((n, l, m_l, m_s) = (3, 2, m_{l2}, -\frac{1}{2})\), where \(m_{l2}\) is one of \(-2, -1, 0, 1, 2\) and \(m_{l2} \neq m_{l1}\)